Symmetry properties of some solutions to some semilinear elliptic equations

In this paper we prove some symmetry results for en- tire solutions to the semilinear equation −∆ u =f (u) , with f non- increasing in a right neighbourhood of the origin. We consid er solutions decaying only in some directions and we give some s uffi- cient conditions for them to be radially symmetric with resp ect to those variables, such as periodicity or the pointwise decay of some derivatives